S T O

453

STO

d n . ln terms of the sines and tangents of the successivevf ' nc hnation ; but in reducing these to numbers, he

V er been led, by the accumulation of small errors, in that&n/'° perose wa y of proceeding, to give erroneous results;' nt ° * be s ‘ n g u lar mistake of conceiving, that the reallesion of these values was only an approximation.

weight of the section C may be determined in thee Way as the foregoing. But surely more simply thus:

IV

to,

0Ql C draw c S parallel to w z, that is, at right angles to

an< ^ make it equal to wz + n 'b; draw s c at right^ es to to, meeting the vertical Ce in c, then cc repre-t 0 3 the weight of C. From d draw d t parallel and equalid ^ c> draw t d perpendicular to D o, meeting the vertical d dis the weight of D, and so on successively.

' a y> instead of drawing d t parallel to S c, and T d per-tli c , lcu lar to d o, we may at once draw from s, s d' perpen-(>f t ^ ar to d o, which will cut off for us c d! = d d, the weightof ab e section D. It is of no consequence, although the lines„ 'dment do not all run to the same centre o.

1 ^ tbus we obtain a general construction for all thev trti nS ’ which turns out abundantly simple ; for, upon anydiv,,^ hoe, V e, Figure 9, if cb be taken to represent theSfigig Weight of any section, C, and c t be drawn at right' iti^, , s to c o, and b t at right angles to b o, meeting the other0 ^ ’ then t b represents the pressure against the abutment

fi a , n ^ T C the pressure against o c; and by drawing t dtl) e ® 11 angles to do, tc to eo, &c., we have the weights offress Ccess »ve sections represented by c d, de, &c., and thet e, ^ re °o their lower abutments represented by t d,“ W'

0 th er . tcay carry the same mode of determination to the8 '°t>s Sl< ^ e 0 ant l P ass the vertex of the arch. The divi-'’PtVa, r j' e P resen ti n g the weights of the sections will run5 fiUt^ as along the indefinite line c V. The pressures on the“’leg , e P nta will be determined as before. Should the two

°!j a section be parallel, the perpendiculars through t. such a section, therefore, should

haw them will coincide

c.

weight. But should the two lines of abutmenttowards the lower side, the line expressing the weight

a,

Of

Si, s ection will return upon the vertical, showing thatr og) Section requires the reverse of weight, viz., a supportSib it el °w. The line tv drawn horizontally through t,I 8 8 the horizontal pressure, which is uniform throughSj T e equilibrated arch. But it is evidently greater, the“titig an< l c t are inclined to each other, the weight b c8 ecti° nstant > that is, the smaller the angle of the wedges,> 0n S s - It also increases directly as the weight of theSi), Q S> &c. The line v e expresses the weight of the Semi-tic Perpendicular pressure on each pier; being the sum®'ghts of all the sections in the semi-arch.

S gqb a | n > it is obvious that the angles b t c, or c t d, &c.,to the angles of the sections boc, cod, &c. If,61 te qu ! -' le we '"bt of any section, E, be given = de, and| i 8 . 8 Wj n ^ Sl . te angle of that section be required, everythingoN known, we have only to join t e, and the line e o,i ^ hie 1 q Wi1 P er pendicular to t e, will exhibit the inclination8 e C fi Vv ’ er abutment of the section ; rf t e is the angle of1 i 8> b ° n - And here it matters not where the point e be,

t^'Sht' ; 3 ° w 8 r eat the base of the section be, provided thet^if, equal to de. We also see, that while the anglese s ame, and the weights proportional, it is of no; tw what the curve passing through the lower edges

i or through their upper edges, may be, they! . e straight lines. According to this principle, the

! ik- > h e 8 not confined to given forms of intrados or extra-^efql . ta ke whatever curve appears most beautifuland, what is more, by the proper adjustment of5 z

the joints, he may cast the ultimate pressure in any direc-tion which he thinks most conducive to the strength ofthe edifice.

“We now proceed to show the application of this investi-gation to some practical cases ; and the first we shall con-sider, is that known by the common, though awkward nameof the flat arch; one with which every mason is perfectlyfamiliar, though it be seldom noticed by writers on equili-bration.

“ Figure 10.—ab ba is a structure of this kind, adjustedto this equilibrium, and resting on the abutments a a, b b.Its construction is exceedingly simple ; nothing more isnecessary than to draw all the joints, m m, l L, &c., to onecentre, c : and the reason is obvious ; for d k, el, &c. arethe differences of the natural tangents of the inclinations ofthe abutments, the perpendicular, CD, being radius : and thesame thing is true in the line d a, and in every other parallelsection. The surface, therefore, Am, ul, that is, the bulksor weights of the stones, are in the same ratio, and it is thatwhich is required by the above principles. Also, if weassume the line of its base to represent the weight of anystone in the arch, for example, kd for half the keystone ;then the perpendicular c d is the horizontal thrust , drift, orshoot of the arch. By increasing D C, or diminishing it, thatis, by drawing the joints to a lower, or a higher centre, wemay alter this thrust at pleasure. What if we should take cup to d ? Some curious ideas occur here, but being chieflyspeculative, we shall not now pursue them. They serve toconnect this case very neatly with the lintel and the Egyptianarch, (or that formed by flat courses of stones gradually over-lapping each other, until the opening be covered), in each ofwhich the horizontal thrust vanishes. We ought also toobserve, that whatever weight of stuff lies on an arch of thiskind, there is no change of design requisite, so long as theupper surface, or roadway, is horizontal. For being every-where of the same height, the mass incumbent on any stonewill be proportional to its base, viz., the back of that stone ;since we must conceive the stuff to press vertically. It istherefore the same as if the whole arch had undergone achange of specific gravity ; every pressure will be increasedin the same proportion.

“ The design of an equilibrated horizontal arch, or plat-band, being thus easily formed, it will not be difficult toextend it to a curve of any form ; abb b'd' a! Figure 10, isan arch of this kind. It is a circular segment from the cen-tre, C, to which the joints of the horizontal arch were directed ;the two key-stones have the same weight and obliquity ofabutrhent ; consequently, the horizontal thrusts are the same.The other arch-stones being previously intended to have thesame weight with those of the flat-arch, it is only necessaryto draw the lines 1 1, 2 2, 3 3, parallel to m m, l Z, k k, andso as to produce this equality.

“ This being merely a simple problem in mensuration, weshall not occupy the reader’s attention with the solution ofit. In the Figure referred to, we have divided the soffit, a b,of the flat arch into equal parts ; all the stones, therefore, ofthat, as well as the curvilinear form, are of equal magnitudeand weight, the angles of the arch-stones only, varying.

“ The reader must have already observed, that when c dexpresses the horizontal thrust, or pressure of the vertex,c m, c k, c l, &c. express the perpendicular pressures on thesuccessive joints m m, k k, 'Ll, &c. Now it is obvious,that K k, L l, &c. are proportional to C K, C L, &c. ; for a d,a d, are parallel. Therefore, the vertical sides of the archbeing parallel, the pressure on each joint of the flat-arch isalways proportional to the surface of that joint, and thepressure on each square inch of joint throughout the arch is